Équipe de Recherche en Physique de l'Information Quantique

Introduction to quantum information Alexandre Blais (2002)

This text exposes the basic properties of quantum information without going into mathematical difficulties as much as possible. Before going to quantum information let us first review the basic and familiar concepts related to ordinary or, as we will now call it, classical information.

Classical Information

The fundamental unit of classical information is the bit (which stands for binary digit). A bit can take two values which are usually denoted 0 and 1. Of course this choice of notation is completely arbitrary and we might have chosen to represent our bits by balls of different colors, say red and green[1]:

A red ball will then stand for a 0 while a green one for a 1. Bits (balls) are precious and we want to protect them as best as we can. To do so, we put our bits in boxes. One box for each ball. For example, imagine we have a red ball that we cared for a great deal. We put it in a box to protect it from any potential problems and unfortunate encounters (we wouldn’t want anyone to paint our red ball in green). The ball is keeped in this safe location until, at a later time, we open the box and take it out. Of course, since a red ball went in, a red ball will come out for certain :

This is a basic property of classical information : reading classical information doesn’t change or perturb in any way its value (of course!). As a result, classical information (balls) are easily copied. Moreover, classical information can be efficiently protected from errors.

Quantum Information

We have seen that the basic unit of classical information is the bit and that we can represent bits by balls of two different colors. Of course, bits a really useful (without them you wouldn’t be able to read this text) but they are not really surprising or exciting by themselves. Let us now consider what would be the properties of information if it was to follow the strange rules of quantum mechanics [2].

We will call the fundamental unit of quan tum information a quantum bit or qubit for short. As for it’s classical counterpart, a qubit can take two values that we again represent by red and green balls. This is however no ordinary ball, it is a quball!Qubit are also very precious and we want to protect them from any trouble. To do so we will do as in the classical case and our put quballs in boxes. There is a difference here however: to take into account the strange properties of quballs we need boxes with two doors:

As in the classical case, if we put a red quball in door #1 and open this same door at a later time, what we see is a red quball (the same would be true for a green quball). In the same way, if we put a red (green) quball in door #2 and open this same door at a later time, we also see a red (green) quball:

Nothing surprising or even interesting so far. But this is not the whole story...

Quantum Strangeness

Let’s now try something new that was not possible in the classical case when we had only one door. First put a red quball in door #1 and, rather than opening this same door, we now open door #2:

In this case, we will not necessarily see a red quball. We see either a red or a green quball! In fact, if we where to repeat this same exercise many times we would observe that in 50% of the cases, we get a red quball and in the other 50%, a green one. The result is thus completely random and does not depend on the color of the quball we used in the first place (in this case red).

This is strange and so let’s sum up what we have learned so far: if we put a quball in a door and open this same door at a later time, we see a quball of the color we had put in. However, if we put a quball through a certain door (say #1) and open the other one (here #2) then the result can be either red or green. This is a surprising finding and it begs for an explanation.

The interpretation physicists have of this phenomena is that when a quball is prepared in a given way (through a certain door) before we look at it by this same door, the quball is just as we had prepared it in the first place. Then when we look at it by this same door the result will be what we expect: the color we had used the first place. Well, this is not profound or anything...

However, if we prepare our quball through a given door (say #1), before we look at it from the other door (that is, from the point of view of the other door; here #2) our quball is both red and green at the same time! As a res ult when we open this door (here #2) we get either red or green each with a probability 1/2. Physicist like to call this weird situation when a quball is simultaneously red and green a quantum superposition: we say that the ball is in a superposition of red and green.

Moreover, after opening the box and seeing say a green ball, then the quball is no longer in a weird superposition of states, it is a gre en quball and that is all. The act of opening the box and looking at the quball has perturbed its state, it has destroyed the superposition! [3] There is no analogy for this for regular, classical, balls. (The information, (i.e. the bits) of which this text is made of is not affected by the action of someone looking at it!)

You are probably not satisfied with the previous ’explanation’ of quantum superposition and the disrupting effect of measurements. Nothing has been said of why, in a given experiment, we get a red quball and in a second experiment a green ball when the preparation was the same in both cases. In fact, could we predict for any sequence of preparation and measurement what will be the color of the quball? The unfortunate answer to those questions is that there is just no explanation for this. This strange behavior of quballs is all that quantum mechanics tell us. The quantum theory doesn’t give us the tools to predict what will be the result of any given measurement, it only give us access to probabilities : that is quantum mechanics can help us predict the statistics of many measurement but cannot tell anything more about a given measurement. Moreover, it doesn’t provide us with much insights on what is the actual meaning of a quantum superposition, of what it means for a ball to be green and red at the same time.

What we know however is that when we apply the strange rules of quantum mechanics (the superpositions and probabilities) to explain what we observe of nature, we get very good results. In fact quantum theory is the theory that as so far the most success compared to any other physical theory. This should be a good reason to accept those strange rules and keep on looking at what are the consequences on information and its manipulation.

A Consequence : No-Cloning

Let us now look at a first consequence of those strange rules: the impossibility of copying quantum information. To do so, imagine the following game: Alice prepares a quball without showing or telling Bob how. Then, she gives him the box containing her precious quball and ask him to copy it. If he succeeds, he wins.

For example, consider the case where Alice puts a green ball through door #2 and then give the closed box to Bob (Bob receives the box indicated by a question mark):

To create of a copy, Bob must know the state (color) of the quball. To obtain this information, he must open the box and look inside. If he opens door #2, he sees a green quball and can go on make a copy : he wins the game. However, if he opens door #1, he sees a quball of the wrong color with probability 1/2 and then produces an incorrect copy. We see that with some probability, the copy operation is unsuccessful.

This result is stronger when we realize that, without Alice’s help, Bob couldn’t tell if his copy was conform to the original or not simply because he has no way of knowing if he opened the right door or not. Hence, since we cannot copy what we don’t know, we conclude that it is impossible to copy quantum information with a high probability of success.

For some applications (like quantum error correction where we would like to make many copies of a qubit to protect it) this property of quantum information can be very annoying. In other cases, like quantum cryptography, this proves very useful.

Quantum Operations

As we know, bits are all that matters for computers. Bits are what is used to store information on your hard drive, it is bits that are processed by your CPU went you execute softwares, its bits that are decoded and processed by your graphic card to show you this text on your screen (if you read it online)...

Imagine now a computer that uses not bits but qubits: a quantum computer. Would such a beast be useful? Could it be more powerful than a computer using regular bits? The answer to both of those questions is yes. Before seeing why it is so, let us first explain how we can manipulate qubits, that is, apply logical operations on them.

The aim of logical operations is to change the value of quballs in a controlled way. One interesting operation on quballs is to create superpositions. How can this be realized? We know that if a quball is prepared through door #1 then, before opening any doors, from the point of view of door #2 the quball is in a superposition of state: if we were to open door #2, we would obtain either red or green. We thus know how to prepare superpositions.

But imagine we had access only to the door which is on top of the box. How would we proceed? A simple way would be to prepare a quball from door #1 (which is initially on top) and then rotate the box such that door #2 now lies on top (this is a rotation of the box by 90°). We then have a superposition from the point of view of the top door :

In this drawing, the quball on the left represent the initial state and the quballs on the right the final state. The diagram in between represent the operation that is applied on the initial quball. Here it is a rotation as is represented by the circular arrow.

This rotation may seem unnecessary but it is not the case: in reality for some very technical reasons, we would only have access to the top door in actual quantum computers. This rotation is then similar to what would actually be used by such a computer and this is what we will use from now on in this text to prepare superpositions.

Let us now describe an operation, or gate, that is very useful in the manipulation of quantum information: the Controlled-Not (or CNOT). As opposed to the previous rotation operation, this gate acts on two quballs:

As before,the quballs on the left is the initial state and on the right is the re sult of the application of the CNOT. The CNOT is represented graphically by the vertical line starting with a circle and ending with a x. Since we will only be considering the top door from now on, we might as well omit to draw the box. We can th en redraw the CNOT as:

Using this notation, the action of the controlled-Not on the four possible color arrangements of 2 quballs is given in the following table:

CNOT : Truth Table

INPUT

OUTPUT

This gate’s action is to invert the second quball (bottom line in the diagram and rightmost quballs in each columns of the table) only if the first one is green.

Quantum Networks

It is useful to put different quantum gates together to realize more complex operations. Consider for example the follow ing network with two red quballs as initial input:

The first step, a rotation by 90°, has for effect to create a superposition of state of the first quballs. The two quballs are then in the state:

That is, the first quball is simultaneously red and green (a superposition of colors) while the second is with certitude red (after all, no operation was applied to the second quball yet).

Now apply the CNOT on the previous two quball state. As can be seen from the truth table corresponding to this operation, the combination red-red is left unchanged while green-red becomes green-green. The state is then:

1/2

1/2

The output of this network is a special superposition of states, one involving two qubits rather than only one as in the previo us examples. For this superposition, the quballs are simultaneously both red and both green! This is quite strange but this is what is obtained by applying the rules of quantum mechanics.

Now, what are the possible results of measurement. If we follow what was said earlier, the two possible outcomes are either two red quballs or two green ones. This means that, after measurement, if we look at one quball only we automatically know the color of the other: it will be the same as the first one.

The fact that the quballs are always of the same colors after going through this network is quite extraordinary. It is so special and interesting that we give a particular name to this situation, we say that the quballs are entangled.

Entanglement

Let’s now see why the two quballs state that was prepared in the previous section is so particular that it was worth giving it a special name.

Imagine two initially red quballs shared between Alice and Bob : Alice has the first one while Bob owns the second one. Together, they apply the network described in the last section. As discussed previously, their quballs is then in the superposition of both quballs being red and both quballs being green. After this preparation, Alice stays in her lab on earth while Bob goes on a space cruise to the Andromeda galaxy.

Once he is there, Alice decides to measure her quball and the result is red. From the discussion above, immediately Bob’s quball is now also red: if Bob was to measure is quball, he would obtain a red quball with certitude. What appended here? An action made on earth by Alice had an immediate reaction a few light years away on Andromeda. It then seems that a signal went instantaneously from Alice’s quball to Bob’s quball to inform it of the measurement result and tell it what color to take from now on. This is quite strange because according to the theory of special relativity developed by Einstein in 1905 [4], no information can propagate at a speed which is greater then the speed of light (if the signal went instantaneously from earth to Andromeda it certainly was going faster than the speed of light).

This result is so puzzling that in 1935, Einstein together with Podolsky and Rosen, argued using a scenario similar in spirit to the one above that, if quantum mechanics predicts such strange and ’unnatural’ behaviors, it must certainly be incomplete and so cannot describe nature properly. This has become known as the EPR paradox [5].

Fortunately for quantum computers and unfortunately for Einstein and his friends, experiments have proven that entanglement exist and does show the strange behavior described here [6]. It then seems that quantum mechanics, as we argued before, is a good description of nature. It is also important to note that the special theory of relativity is not violated in the above situation because there was no information exchange between the two quballs: there is no way Alice and Bob could use the two quballs to exchange information.

Even though there is no information exchange possible using entangled pairs of quballs (we will also call this an EPR pair) alone, those states are very useful. It is EPR pairs that renders quantum computers powerful (as we will see later). EPR pairs also makes possible quantum teleportation.

Quantum Teleportation

’’Beam me up, Scotty’’. -Captain Kirk

Imagine Alice would like to give a particular quball to Bob (which is still on Andromeda). Unfortunately, someone else prepared this quball for her and she doesn’t know how he did. As a result, she cannot tell Bob how to prepare an identical quball just by talking to him on the intergalaxial phone. Also, this quball is very precious and she doesn’t have any confidence in the intergalaxial post service (of course the real problem here is that the custom officers will insist in opening the box, thereby ruining her quball...).

Fortunately, they still have an unperturbed EPR pair left from the last time they met. Using this pair she will be able to send Bob her quball using quantum teleportation. Let’s now describe how this is possible.

This is probably the most technical part of this text, as we will now bring together everything we have learned so far and see how those weird rules of quantum theory can actually be useful. So let’s start by the beginning and go slowly. First, Alice’s quball can be in any state : it can be red, green or both. To be as general as possible, let’s represent it in the following superposition:

c

d

This is very general because setting c=1 and d=0 is like saying that the quball is only red (there is no green part). Alternatively, if c=0 and d=1 this means that it is green. The last possibility we have already seen is c=d =1/2, meaning that it is both red and green simultaneously, this represent a superposition of both colors. To make very clear that this quball is own by Alice, we added the label ’A’ on it (a ’B’ would mean that it is own by Bob).

Now, as we have said, Alice and Bob must also share an EPR pair:

1/2

1/2

As seen from the labels, the first quball is owned by Alice and the second by Bob. Something new in this example of application is that we are dealing not only with one or two quballs but three (the quball Alice wants to send and the two quballs in the entangled pair) and we now have to write the state describing completely this system. The way to do this follows from what was said in the section ’Quantum Networks’. There are two possibilities for the first quball and two for the EPR pair. This means that there is a total of 2x2=4 possibilities for the three quballs:

where the factors 1/2 were not explicitly written (the horizontal lines are just guides for the eyes). The three quballs are in a superposition of 4 different colors arrangements.

Now, Alice performs a Controlled-Not on her quballs. Applying the rules given earlier in the ’Truth Table’ of this operation (change the color of the second only if the first one is green), the result is

c

d

c

d

Of course, only Alice’s quballs have been affected by this operation. Then, Alice applies a rotation of 90° on her first quball (left most in each line). Now it will be a little messy because, as we have seen before, such a rotation brings a quball which was in a given color to a superposition of both possible colors. We had a superposition of 4 states and the result is a superposition of 2x4=8 states because the first quball (left most) in each line will now either be red or green:

Fortunately, we are almost done. What is essential to notice in order to complete the transfer is that this superposition of eight different color arrangements has a special structure: there are two arrangements (lines) where Alice’s quballs are both red (1st and 7th line), two where they are respectively green and red (line 2 and 8), two where they are respectively red and green (line 3 and 5) and finally two where both of Alice’s quballs are green (line 4 and 6). To make this fact more explicit, let’s rearrange the rows of this superposition such as to make those pairs of lines neighbors:

From the right hand side of the equality, we notice that, corresponding to each of the four possible color arrangements for both of Alice’s quballs, corresponds a superposition of Bob’s quball. When Alice’s quballs are both red or green-red then, Bob’s quball is exactly in the same state as the quball Alice wanted to send him! However, when Alice’s quball are red-green or both green then, Bob’s quball is almost right: only the coefficients ’c’ and ’d’ are mixed up (this is emphasized on the image with these coefficients being of a different color). This correspondence between Alice’s and Bob’s quballs is a manifestation of the fact that those three quballs are now entangled [7].

To complete the procedure, Alice measures her quballs. There are four possible results: the four color arrangements of two quballs listed on the previous image. Of course, since Alice’s part is entangled to Bob’s part, this means that after this measurement, Bob’s quball will be in the state corresponding to Alice results (see the image).

As we stated earlier, in two of the possible measurement outcomes, Bob’s qubit is in the state Alice wanted to send him! Unfortunately, in the two other possible measurement outcomes, the state is not the same: the coefficients are mixed up. To settle this situation, Alice tells Bob what is the color she measured for her two quballs. Bob then knows if the coefficients are mixed or not. If they are not mixed up, the procedure is complete. If they are mixed up, then, he can change a red quball by a green quball and vice versa (pretty much like a controlled-not can change a red quball by a green quball). Teleportation is now complete: Alice has succeeded in ’giving’ Bob a quball in exactly the state she wanted without ever sending him this particular quball. This is why this procedure is called teleportation.

Notice that to complete the transfer, Alice had to call Bob on the intergalaxial phone and tell him the result of her last measurement. Without this call, Bob would have no way of knowing if his quball is right or not. As a result, this whole procedure would be useless. Of course, talking on the intergalaxial phone takes a long time (remember that information cannot travel faster than the speed of light) this means that teleportation, as a whole, cannot be executed instantaneously. Einstein would be happy about this conclusion since it means that special relativity is not violated by quantum teleportation (the information about Alice’s quball did not travel faster than the speed of light since to complete the transfer Bob had to wait for Alice’s phone call to be completed).

Quantum teleportation is not just a theorist dream but has actually been realized experimentally, using photons and nuclear spins instead of balls however [8]. The application of this technique to larger objects, say human beings, is however still a far way ahead and very unlikely to ever be realized.

Classical Computers

One of the things we have learned so far is how quballs are different from regular balls, how quantum information differs from classical information. One of the motivations to study those differences was the question of whether a computer following the rules of quantum mechanics could be useful and perhaps even more powerful than regular classical computers. It is now time to answers this question but, as before, let us start by looking at the more familiar case.

Imagine a classical computer (like a PC) but made out of only 7 bits. As before, those 7 bits can be represented by balls. There is many possible arrangements of those 7 balls:

There is a total of 2x2x2x2x2x2x2 = 2^7 = 128 such possible arrangements for 7 bits. Since a bit can only be red or green (and not simultaneously both colors) then the state of the classical computer is very easy to describe: each of those 128 arrangements are mutually exclusive and all is needed to represent the complete state of the computer is 7 bits (of course, we had a 7 bit computer so this is not very difficult to understand!). This was a very simple observation but will nevertheless be useful, to understand the differences with the quantum case.

Quantum Computers

A 7 quball (qubit) quantum computer

Imagine now a quantum computer made out of only 7 qubits. As in the classical case, there is 2^7=128 possible arrangements of 7 quballs. Here, the description is not simple at all. In this quantum case, we cannot say that a given quball is red or green because it as the potential to be either red or green depending on which door is opened. Moreover, because quballs can be entangled with other quballs, opening one box can have an impact on what is obtained when the other boxes are opened (remember that for an EPR pair, if we open one box and find red, then we know even before opening the other box that the result will be red; our first measurement as had an influence on the result of the subsequent measurement). A complete description of the 7 quballs needs to take into account those extra subtleties that were of no concern for classical information.

It turns out that to describe completely the 7 quballs, not only 7 numbers or regular balls are needed as in the classical case but, we rather need 2^7 regular classical balls (one for each possible arrangements). This is exponentially more than in the classical case! This makes a huge difference as can be seen from the following table:

# of quballs

# classical balls required for a complete description

10

1024

20

1 048 580

20

1 0 73 470 000

300

Plus que le nombre d’atomes dans l’univers visible !

As can be seen from this table, the number of classical balls needed for describing correctly quantum information is humongus. For example, for 300 quballs (a number which is quite modest compared with the billions of transistors in a modern PC) the number of classical balls required is greater than the number of atoms in the visible universe! This means that if we were to take all the matter in the visible universe, we would still not have enough resources to describe completely those 300 quballs! Of course, this means that the classical simulation of a mere 300 quballs, not saying anything of larger numbers of quballs, is hopeless...

Since a classical computer (using regular classical bits) is not able to simulate efficiently a quantum computer and since a quantum computer (using qubits) is able to simulate efficiently another quantum computer (yet another obvious statement) then the quantum computer must be, at least in some sense, more powerful than its classical counterpart. This is the conclusion at which Richard Feynman arrived in the earl y 1980’s and which sparked the interest in this field.

Going beyond balls

Balls and quballs are an easy way (hopefully!) to understand the basic ideas related to classical and quantum information. Using balls, we were able to show the basic properties of those two kinds of information and their principal differences. We have seen how quantum information behaves strangely and particularly when we make preparation using one door and reading out from the other door we get a random result. Using this notation we were able to obtain some important results like the no-cloning theorem and useful applications like quantum teleportation.

However, as became clear in the discussion related to quantum teleportation, this notation is a bit clumsy when dealing with large numbers of balls. But now that we understand the basic ideas related to quantum information, a much more useful notation is to represent balls (or quballs) by something simpler: say 0s and 1s! After all, it is easier to write a 0 or a 1 than to draw red and green balls.

Coming back to the 0 and 1 we have come full circle and are now ready to use this simpler notation and go further in our exploration of quantum information. Of course, this deeper exploration leads us beyond this introductory text but there are many resources that can serve as guides in this new and innovative field of research. Apart from the references already cited [1-8] you should continue to explore this web page to find out more about algorithms and physical implementations (this tries to answer the question of how exactly to realize quballs in the lab).

[1]The idea of using balls of different colors to represent info rmation is due to John Preskill from Caltech University. For some of his papers using this simple notation: 1) Battling Decoherence: The Fault-Tolerant Quantum Computer, Physics Today, p24, June 1999. 2) Robust Solutions to Hard Problems, Nature, p631, February 1998. Also, check out his web page where you will find transcripts of some of his lectures (some of the examples in the present text were inspired from those t ranscripts) and lecture notes for a graduate course on quantum information.

[2] For a fun to read and accessible to all introduction to quantum mechanics a good place to start is Richard Feynman book’s ’’QED : The Strange Theory of Light and Matter’’. Princeton University Press, 1985.

[3]The fact that measurement of a superposition perturbs the state and renders observation of the superposition impossible shouldn’t be to surprising. After all, have you ever seen such a weird superposition of states? As a result, the theory we have for quballs shouldn’t predict the possibility of observing such states as it would go against our everyday experience.

[4]Albert Eisntein, The Meaning of Relativity. Princeton University Press, 1966.

[5] For more details on entanglement and the EPR paradox, see this webpage.

[6] The experiments were not done using quballs but rather with photons, particles of light. A group of researchers entangled two photons, separated them by 10km using optic fiber and then measurement them. They observed the same strong correlations as we described between the quballs. Here is the reference to the original paper: Gisin et al. Physical Review Letters, 81 3563 (1998). You can get this paper from this web page.

[7]Remember that this is exactly what we had for the EPR pair: to a particular color of the first quball correspond a particular color for the second one (they were always both the same). In this case, the correspondence is different since there is three instead of two quballs.

[8] For more information on teleportation and particularly on experimental realisations, a good start is this Scientific American article. The following webpage is also instructive.